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SageMath
E = EllipticCurve("ci1")
E.isogeny_class()
Elliptic curves in class 33810ci
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33810.ck7 | 33810ci1 | \([1, 1, 1, -151054065, 712625029647]\) | \(3239908336204082689644289/9880281924658790400\) | \(1162405288154182031769600\) | \([4]\) | \(7962624\) | \(3.4862\) | \(\Gamma_0(N)\)-optimal |
| 33810.ck6 | 33810ci2 | \([1, 1, 1, -215279345, 47996142095]\) | \(9378698233516887309850369/5418996968417034240000\) | \(637539574337295661301760000\) | \([2, 2]\) | \(15925248\) | \(3.8328\) | |
| 33810.ck3 | 33810ci3 | \([1, 1, 1, -12226410225, 520346103390735]\) | \(1718036403880129446396978632449/49057344000000\) | \(5771547464256000000\) | \([4]\) | \(23887872\) | \(4.0355\) | |
| 33810.ck8 | 33810ci4 | \([1, 1, 1, 860368655, 384889095695]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-40823833787702621021212761600\) | \([2]\) | \(31850496\) | \(4.1794\) | |
| 33810.ck5 | 33810ci5 | \([1, 1, 1, -2318531825, -42823861109233]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(5460722159232899676600000000\) | \([2]\) | \(31850496\) | \(4.1794\) | |
| 33810.ck2 | 33810ci6 | \([1, 1, 1, -12226425905, 520344701981327]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(1080081136193597015625000000\) | \([2, 2]\) | \(47775744\) | \(4.3821\) | |
| 33810.ck4 | 33810ci7 | \([1, 1, 1, -12012050905, 539470982231327]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-14801317104311412612090622875000\) | \([2]\) | \(95551488\) | \(4.7287\) | |
| 33810.ck1 | 33810ci8 | \([1, 1, 1, -12441051785, 501128731842815]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(14749473186133861541748046875000\) | \([2]\) | \(95551488\) | \(4.7287\) |
Rank
sage: E.rank()
The elliptic curves in class 33810ci have rank \(0\).
Complex multiplication
The elliptic curves in class 33810ci do not have complex multiplication.Modular form 33810.2.a.ci
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
